On the Existence of Hamiltonian Stationary Lagrangian Submanifolds in Symplectic Manifolds

Let (M,ω) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with ω. For instance, g could be Kähler, with Kähler form ω. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional Volg under Hamiltonian deformations, computing Volg (L) using g|L. It is called Hamiltonian stable if in addition the second variation of Volg under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in Cn which is Hamiltonian rigid, then for any M,ω, g as above there exist compact Hamiltonian stationary Lagrangians L′ in M contained in a small ball about some p ∈ M and locally modelled on tL for small t > 0, identifying M near p with Cn near 0. If L is Hamiltonian stable, we can take L′ to be Hamiltonian stable. Applying this to known examples L in Cn shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to Tn, and to (S1 × Sn−1)/Z2, and with other topologies, in every compact symplectic 2n-manifold (M,ω) with compatible metric g.